Posts Tagged ‘mechanical power transmission’

The Two T’s of the Euler-Eytelwein Formula

Sunday, May 14th, 2017

    Last time we introduced some of the variables in the Euler-Eytelwein Formula, an equation used to examine the amount of friction present in pulley-belt assemblies.   Today we’ll explore its two tension-denoting variables, T1 and T2.

   Here again is the Euler-Eytelwin Formula,where, T1 and  T2 are belt tensions on either side of a pulley,

T1 = T2 × eθ)

    T1 is known as the tight side tension of the assembly because, as its name implies, the side of the belt containing this tension is tight, and that is so due to its role in transmitting mechanical power between the driving and driven pulleys.   T2 is the slack side tension because on its side of the pulley no mechanical power is transmitted, therefore it’s slack–it’s just going along for the ride between the driving and driven pulleys.

   Due to these different roles, the tension in T1 is greater than it is in T2.

The Two T’s of the Euler-Eytelwein Formula

The Two T’s of the Euler-Eytelwein Formula

    In the illustration above, tension forces T1 and T2 are shown moving in the same direction, because the force that keeps the belt taught around the pulley moves outward and away from the center of the pulley.

    According to the Euler-Eytelwein Formula, T1 is equal to a combination of factors:  tension T2 ; the friction that exists between the belt and pulley, denoted as μ; and how much of the belt is in contact with the pulley, namely θ.

    We’ll get into those remaining variables next time.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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A First Look at the Euler-Eytelwein Formula

Friday, May 5th, 2017

    Last time we introduced the Pulley Speed Ratio Formula, a Formula which assumes a certain amount of friction in a pulley-belt assembly in order to work.   Today we’ll introduce another Formula, one which oversees how friction comes into play between belts and pulleys, the Euler-Eytelwein Formula.   It’s a Formula developed by two pioneers of engineering introduced in an earlier blog, Leonhard Euler and Johann Albert Eytelwein.

    Here again is the Pulley Speed Ratio Formula,

D1 × N1 = D2 × N2

where, D1 is the diameter of the driving pulley and D2 the diameter of the driven pulley. The pulleys’ rotational speeds are represented by N1 and N2.

   This equation works when it operates under the assumption that friction between the belt and pulleys is, like Goldilock’s preferred bed, “just so.”   Meaning, friction present is high enough so the belt doesn’t slip, yet loose enough so as not to bring the performance of a rotating piece of machinery to a grinding halt.

    Ideally, you want no slippage between belt and pulleys, but the only way for that to happen is if you have perfect friction between their surfaces—something that will never happen because there’s always some degree of slippage.   So how do we design a pulley-belt system to maximize friction and minimize slip?

    Before we get into that, we must first gain an understanding of how friction comes into play between belts and pulleys.   To do so we’ll use the famous Euler-Eytelwein Formula, shown here,

A First Look at the Euler-Eytelwein Formula

A First Look at the Euler-Eytelwein Formula

   

where, T1 and  T2 are belt tensions on either side of a pulley.

    We’ll continue our exploration of the Euler-Eytelwein Formula next time when we discuss the significance of its two sources of tension.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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A Pulley Speed Ratio Formula Application

Friday, April 21st, 2017

    Last time we saw how pulley diameter governs speed in engineering scenarios which make use of a belt and pulley system.   Today we’ll see how this phenomenon is defined mathematically through application of the Pulley Speed Ratio Formula, which enables precise pulley diameters to be calculated to achieve specific rotational speeds.   Today we’ll apply this Formula to a scenario involving a building’s ventilating system.

    The Pulley Speed Ratio Formula is,

                                                D1 × N1 =  D2 × N2                             (1)

where, D1 is the diameter of the driving pulley and D2 the diameter of the driven pulley.

A Pulley Speed Ratio Formula Application

A Pulley Speed Ratio Formula Application

   

The pulleys’ rotational speeds are represented by N1 and N2,  and are measured in revolutions per minute (RPM).

    Now, let’s apply Equation (1) to an example in which a blower must deliver a specific air flow to a building’s ventilating system.   This is accomplished by manipulating the ratios between the driven pulley’s diameter, D2, with respect to the driving pulley’s diameter, D1.   If you’ll recall from our discussion last time, when both the driving and driven  pulleys have the same diameter, the entire assembly moves at the same speed, and this would be bad for our scenario.

    An electric motor and blower impeller moving at the same speed is problematic because electric motors are designed to spin at much faster speeds than typical blower impellers in order to produce desired air flow.   If their pulleys’ diameters were the same size, it would result in an improperly working ventilating system in which air passes through the furnace heat exchanger and air conditioner cooling coils far too quickly to do an efficient job of heating or cooling.

    To bear this out, let’s suppose we have an electric motor turning at a fixed speed of 3600 RPM and a belt-driven blower with an impeller that must turn at 1500 RPM to deliver the required air flow according to the blower manufacturer’s data sheet.   The motor shaft is fitted with a pulley 3 inches in diameter.   What pulley diameter do we need for the blower to turn at the manufacturer’s required 1500 RPM?

    In this example known variables are D1 = 3 inches, N1 = 3600 RPM, and N2 = 1500 RPM.   The diameter D2 is unknown.   Inserting the known values into equation (1), we can solve for D2,

                               (3 inches) × (3600 RPM) = D2 × (1500 RPM)         (2)

Simplified, this becomes,

                                            D2 = 7.2 inches                                      (3)

    Next time we’ll see how friction affects our scenario.

   

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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The Difference Between Driven and Driving Pulleys

Friday, March 31st, 2017

    Last time we introduced two historical legends in the field of engineering who pioneered the science of mechanical power transmission using belts and pulleysLeonhard Euler and Johann Albert Eytelwein.   Today we’ll build a foundation for understanding their famous Euler-Eytelwein Formula through our example of a simple mechanical power transmission system consisting of two pulleys and a belt, and in so doing demonstrate the difference between driven and driving pulleys.

    Our example of a basic mechanical power transmission system consists of two pulleys connected by a drive belt.   The driving pulley is attached to a source of mechanical power, for example, the shaft of an electric motor.   The driven pulley, which is attached to the shaft of a piece of rotating machinery, receives the mechanical power from the electric motor so the machinery can perform its function.

The Difference Between Driven and Driving Pulleys

The Difference Between Driven and Driving Pulleys 

   

    Next time we’ll see how driven pulleys can be made to spin at different speeds from the driving pulley, enabling different modes of operation in mechanical devices.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Johann Albert Eytelwein, Engineering Trailblazer

Monday, March 20th, 2017

    They say necessity is the mother of invention, and today’s look at an influential historical figure in engineering bears that out.   Last week we introduced Leonhard Euler and touched on his influence to the science of pulleys.   Today we’ll introduce his contemporary and partner in science, Johann Albert Eytelwein, a German mathematician and visionary, a true engineering trailblazer whose contributions to the blossoming discipline of engineering led to later studies with pulleys.

 Johann Albert Eytelwein, Engineering Trailblazer

Johann Albert Eytelwein, Engineering Trailblazer

   

    Johann Albert Eytelwein’s experience as a civil engineer in charge of the dikes of former Prussia led him to develop a series of practical mathematical problems that would enable his subordinates to operate more effectively within their government positions.   He was a trailblazer in the field of applied mechanics and their application to physical structures, such as the dikes he oversaw, and later to machinery.   He was instrumental in the founding of Germany’s first university level engineering school in 1799, the Berlin Bauakademie, and served as director there while lecturing on many developing engineering disciplines of the time, including machine design and hydraulics.   He went on to publish in 1801 one of the most influential engineering books of his time, entitled Handbuch der Mechanik (Handbook of the Mechanic), a seminal work which combined what had previously been mere engineering theory into a means of practical application.

    Later, in 1808, Eytelwein expanded upon this work with his Handbuch der Statik fester Koerper (Handbook of Statics of Fixed Bodies), which expanded upon the work of Euler.   In it he discusses friction and the use of pulleys in mechanical design.  It’s within this book that the famous Euler-Eytelwein Formula first appears, a formula Eytelwein derived in conjunction with Euler.   The formula delves into the usage of belts with pulleys and examines the tension interplay between them.

    More on this fundamental foundation to the discipline of engineering next time, with a specific focus on pulleys.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Leonhard Euler, a Historical Figure in Pulleys

Thursday, March 9th, 2017

    Last time we ended our blog series on pulleys and their application within engineering as aids to lifting.   Today we’ll embark on a new focus series, pulleys used in mechanical devices.   We begin with some history, a peek at Swiss scientist and mathematician Leonhard Euler, a historical figure credited to be perhaps the greatest mathematician of the 18th Century.

   

Leonhard Euler, a Historical Figure in Pulleys

Leonhard Euler, a Historical Figure in Pulleys

   

    Euler is so important to math, he actually has two numbers named after him.   One is known simply as Euler’s Number, 2.7182, most often notated as e, the other Euler’s Constant, 0.57721, notated γ, which is a Greek symbol called gamma.   In fact, he developed most math notations still in use today, including the infamous function notation, f(x), which no student of elementary algebra can escape becoming intimately familiar with.

    Euler authored his first theoretical essays on the science and mathematics of pulleys after experimenting with combining them with belts in order to transmit mechanical power.   His theoretical work became the foundation of the formal science of designing pulley and belt drive systems.   And together with German engineer Johann Albert Eytelwein, Euler is credited with a key formula regarding pulley-belt drives, the Euler-Eytelwein Formula, still in use today, and which we’ll be talking about in depth later in this blog series.

    We’ll talk more about Eytelwein, another important historical figure who worked with pulleys, next time.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Mechanical Power Transmission – Putting the Centrifugal Clutch Together

Sunday, April 29th, 2012

     I’ve never been one to enjoy table top puzzles, yet I love to examine the way mechanical things fit together.  Manipulating parts to see how they interrelate to form an operational system is a pastime I very much enjoy.  In fact, I spend many evenings at my work bench doing just this.  I often become so engrossed in the activity I forget what time it is.  The result is yet another night without TV.  So sad…

     Last week we looked at how a centrifugal clutch mechanism operates when it’s coupled to a gasoline engine shaft spinning at idle speed, and then we depressed the engine throttle trigger to speed things up.  Let’s now introduce a new component called the clutch housing to see how it interfaces with the clutch mechanism to drive the cutter head in a grass trimmer.

centrifugal clutch housing

Figure 1

 

     The clutch housing shown in Figure 1 resembles a rather short cup.  One end is open, the other closed.

     Figure 2 shows the closed end of the clutch housing connected to the cutter shaft’s coupling.  On the cutter shaft coupling resides a ball bearing which enables the clutch housing to both spin freely and act as a support for the clutch housing.  The open end of the clutch housing allows the clutch mechanism to fit neatly inside.

centrifugal clutch assembly

Figure 2

 

     Next time we’ll put the assembly shown in Figure 2 into operation.  First we’ll examine how the centrifugal clutch mechanism and clutch housing operate with the engine at idle speed, then compare that to the engine operating at actual cutting speed.

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Mechanical Power Transmission – The Centrifugal Clutch in Operation

Sunday, April 22nd, 2012
     Just the other day I unexpectedly experienced the effects of centrifugal force while  driving home from the grocery store.  The checker had packed my entire order into one bag, making it top heavy.  Then en route someone cut me off at an intersection, and I had to make a sharp turn to avoid a crash.  During this maneuver centrifugal force came into play, forcing my grocery bag out of its centered position on the front seat next to me.  It lurched into the passenger’s door, fell over, and spilled its contents onto the floor.  Fortunately the eggs didn’t get smashed.

     In previous articles we identified the component parts of a centrifugal clutch mechanism and learned how centrifugal force makes objects spinning in a circular path about a fixed point move outward.  We can now explore what happens when we couple a centrifugal clutch mechanism to the engine of a grass trimmer.

     Figure 1 depicts the spinning clutch mechanism of a gas engine when it’s just been started and is operating at a slow idle speed.

centrifugal clutch mechanism

Figure 1

 

     Like the red ball in my previous article on centrifugal force, the blue centrifugal clutch shoes each have a mass m.  They spin around a fixed point P, situated at the center of the yellow engine shaft coupling.  Point P is located a distance r from the center of each shoe.  The shoes in motion have a tangential velocity V, and in accordance with Sir Isaac Newton’s Law of Centrifugal Force, the force Fc acts upon each shoe, causing them to want to pull out from the center of the mechanism, away from the fixed point.  Since idle speed is rather slow, however, the centrifugal force exerted upon the shoes isn’t strong enough to overcome the tension of the two springs and the coils connecting them remain coiled, holding the shoes tightly in position on the green boss.

     So what happens when we press the throttle trigger on the gas engine and cause the engine to speed up?  See Figure 2.

clutch shoes

Figure 2

 

     Figure 2 shows the clutch mechanism spinning at an increased velocity.  The tangential velocity V increases, and according to Newton’s law, the centrifugal force Fc acting on the clutch shoes increases as well.  The force is so strong that it overcomes the tension in the springs and they extend.  The clutch shoes are caused to move out and away from fixed point P, as well as from each other, traveling along the ends of the boss.

     When we remove our finger from the throttle trigger, the engine will slow down and return to idle speed.  The centrifugal force will decrease and the springs will pull the shoes back towards fixed point P.  The mechanism will return to its previous state, as shown in Figure 1.

     Next time we’ll insert the centrifugal clutch mechanism into the clutch housing to see how mechanical power is transmitted from the engine to the cutter head in our grass trimmer.

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Mechanical Power Transmission – Centrifugal Clutches

Monday, April 2nd, 2012
     I remember the days when trimming grass around trees, fences, and flower beds involved the use of hand operated clippers.  You know, those scissor-like things that require you to squeeze the handles together to move the blades.  Cutting seemed to take forever and there was a lot of bending, stooping, and kneeling which would kill your back and turn your knees green from grass stains.  Worst of all, the repetitive motion of squeezing the handles dozens of times would cramp your hands.  It was a great day when gasoline powered grass trimmers came along.  Just pull the recoil starter cord and you’re ready to go.  It’s fast, easy, and the final result looks better too.

     If you’ve ever operated a gasoline powered tool like a grass trimmer, you probably noticed that the cutter action isn’t immediate once the engine is started.   Instead, the engine enters into a much slower initial speed mode, the idle speed.  The cutter moves only after the throttle trigger is depressed.  This introduces more gas to the engine, causing it to speed up, and this action is due to a device called the centrifugal clutch.

     A centrifugal clutch, or any type of clutch for that matter, serves one basic function, to physically disconnect, then reconnect a gasoline engine from whatever it is powering.  For example, if the engine in a weed trimmer stayed permanently connected to the cutter when the engine was started, it would pose a definite safety hazard.  Even at idle speed, the cutter would immediately kick into high speed operational mode, and if someone wasn’t prepared for this instant response there would be a good probability of injury.

     When a centrifugal clutch is placed between the engine and the cutter, it automatically disconnects the engine from the cutter during starting and at idle speed.  We’ll see how it does that in a later blog.  For now, let’s consider the fact that the idle function serves as a “get ready.”  The user is able to both psychologically and physically prepare themselves to use their tool.  Pressing the trigger revs the engine up and causes the centrifugal clutch to connect the engine to the cutting action.  When the operator takes their finger off the throttle trigger the engine returns to idle speed, and the clutch automatically disconnects the engine from the cutter.  The cutter becomes idle.centrifugal clutch

Figure 1

 

     Figure 1 shows a gas trimmer and its centrifugal clutch.  The engine is on one end of the trimmer and the cutter at the other.  A hollow metal tube runs between them.  This tube  contains the cutter drive shaft.  The centrifugal clutch and its clutch housing are located in a cone shaped compartment between the engine and the metal tube.  The clutch is connected to the engine drive shaft and the clutch housing is connected at the other end of the cutter drive shaft.  When they’re assembled into the grass trimmer, the clutch fits within the clutch housing.

     Next time we’ll see how the centrifugal clutch on a grass trimmer uses centrifugal force and friction to automatically transmit mechanical power from the gas engine to the cutter.

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